Mathematical Modeling of Nonlinear Dynamics of Continuous Mechanical Structures with an Account of Internal and External Temperature Fields

Abstract

This chapter focuses on the construction of mathematical models of nonlinear dynamics of structural members in the form of plates and shallow shells, including internal and external temperature fields. The geometric nonlinearity is taken in the von Karman form, and the physical nonlinearity is introduced based on the strain theory of plasticity, whereas the heat transfer processes are followed with the help of the Fourier principle. The variational formulation yields PDEs of different dimensions and different types (hyperbolic and hyperbolic–parabolic). Our considerations are based on the first-order kinematic Kirchhoff–Love model. The existence of a solution to the coupled problem of thermoelasticity of shells in the mixed form with a parabolic PDE governing heat transfer effects is rigorously proved. The economical (reasonably short computational time) algorithms devoted to the investigation of the coupled problems of the theory of shallow shells with the parabolic heat transfer equation based on the Faedo–Galerkin method in higher approximations and the finite difference method to second-order accuracy have been worked out. In order to solve the stationary problems of the theory of shells, we have extended and modified the classical relaxation method, exhibiting its effectiveness and high accuracy. A wide class of nonlinear vibrations of shells with various types of nonlinearity is studied.